Violation of Virial theorem as indication to ergodicity breaking

Question

Under which conditions the break of virial theorem implies break of ergodicity?

I've seen this question, but it is very limited and not sufficient. To constrain the discussion I'm interested in 1D hamiltonians of the form (2 degrees of freedom)-

$$H=a p^2+bx^2+cx^4$$

That is, confining potential with a steady state. The virial predicts essentially - $$E_k=\frac{1}{2}\langle x\frac{\partial H}{\partial x}\rangle_t = b \langle x^2\rangle_t+2c\langle x^4\rangle_t$$ Here $\langle \cdots \rangle_t$ is time average. Assuming ergodicity, that is (for my purposes) $\langle \cdots\rangle_t = \langle \cdots\rangle$ with $\langle \cdots\rangle$ being ensemble average, we get the prediction that - $$a\langle p^2\rangle=b\langle x^2\rangle+2c \langle x^4\rangle$$

Under which conditions the break of this relation will imply break of ergodicity? Will any violation suffice?

EDIT: I notice it wasn't clear that the system is coupled to some non-thermal bath, which induces stationary drive and dissipation such that the system has time independent steady state, however not Boltzmann-Gibbs. Thus the dynamics are not Newtonian.

Answer 1

I tried to learn a little about the subject, so here is my attempt at an answer. According to van Kampen

https://doi.org/10.1016/0031-8914(71)90105-4

the system is ergodic if and only if all autonomous functions of position in phase space have time averages (for trajectories on an energy surface $E$) equal to phase space surface averages on that surface, for all energies $E$, except for trajectories whose points on the energy surface at some time form a set of zero measure.

Consider the autonomous quantity

$$ F = ap^2 - bx^2 - 2cx^4. $$

According to the virial theorem, which is valid for any energy $E$ thanks to the shape of the potential (it keeps any initial condition to a bounded motion), time average of $F$ is zero. If the equation

$$a\langle p^2\rangle=b\langle x^2\rangle+2c \langle x^4\rangle~~~(*)$$

si violated for some $E$, the space average $\langle F\rangle$ is not zero and therefore the time average of $F$ does not conform to what ergodicity requires, for all points on the energy surface $E$. We have a set of initial conditions with non-zero measure which do not obey the equality of averages, so we conclude that the violation of the equation (*) for space averages is, in this case, sufficient indication of violation of ergodicity.